Evaluate the definite integral. $\int^{2\pi}_{\frac{3\pi}{2}}-4\sin(x)\,dx = $
Solution: First, use the sine rule: $\begin{aligned}\int^{2\pi}_{\frac{3\pi}{2}}-4\sin(x)\,dx~&=~4\cos(x)\Bigg|^{2\pi}_{{\frac{3\pi}{2}}}\end{aligned}$ Second, plug in the limits of integration: $(4\cdot{\cos(2\pi)})-(4\cdot{\cos(\frac{3\pi}{2})}) = 4+0 =4$. The answer: $\int^{2\pi}_{\frac{3\pi}{2}}-4\sin(x)\,dx~=~4$